Contributed Session Fri.2.H 0111

Friday, 13:15 - 14:45 h, Room: H 0111

Cluster 19: PDE-constrained optimization & multi-level/multi-grid methods [...]

PDE-constrained optimization problems with non-smooth structures


Chair: Caroline Löbhard



Friday, 13:15 - 13:40 h, Room: H 0111, Talk 1

Duy Vu Ngoc Luong
A multiresolution algorithm for large scale non-smooth convex optimization problems

Coauthors: Panos Parpas, Daniel Rueckert, Berc Rustem


We develop an algorithm based on multigrid concepts and apply it to nondifferentiable convex optimization problems. The approach utilises the first order method and a hierarchy of models of different fidelity. The goal is to exploit the efficiency of the lower resolution model and propagate the information of the solution to the high resolution model. We discuss the convergence of the algorithm and show numerical results in a Computer Vision application. The convergence results apply to a broad class of non-smooth convex optimization problems.



Friday, 13:45 - 14:10 h, Room: H 0111, Talk 2

Michelle Vallejos
Multigrid methods for elliptic optimal control problems with pointwise mixed control-state constraints


Elliptic optimal control problems with pointwise mixed control-state constraints are considered. To solve the problem numerically, multigrid techniques are implemented. The numerical performance and efficiency of the multigrid strategies are discussed and interpreted in comparison with other existing numerical methods.



Friday, 14:15 - 14:40 h, Room: H 0111, Talk 3

Caroline Löbhard
Optimal control of elliptic variational inequalities: A mesh-adaptive finite element solver

Coauthors: Michael Hintermüller, Ronald H.w. Hoppe


A wide range of optimization problems arise originally in a non-discrete function space setting which has to be discretized in order to find an approximate solution. It is the idea behind mesh-adaption techniques, to find a discrete space that fits best to the unknown continuous solution. While adaptive methods are well-established in solvers for partial differential equations, only a few work has been done for optimal control problems.
We consider the optimal control of an elliptic variational inequality, a problem class with a challenging analytic and algorithmic background on the one hand, and a wide range of applications on the other hand.
Moving on the border line between numerical analysis and computational optimization, we show the principle of goal oriented error estimation operating with the C-stationarity system in the continuous as well as the discrete setting, present a numerical solver for the mathematical program with equilibrium constraints (MPEC) and analyze the benefit of our adaptive solver compared to a method working on a uniformly refined mesh.


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